Simplify the trigonometric expression exactly: (1 − sin A) / (1 + sin A) Write the simplest equivalent form using standard trigonometric identities (assume the denominator is not zero).

Introduction / Context: This question checks identity manipulation and rationalization in trigonometry. The expression (1 − sin A)/(1 + sin A) is commonly simplified by multiplying numerator and denominator by (1 − sin A) to remove the sum in the denominator and convert everything into cos A terms.

Given Data / Assumptions:

• Expression: (1 − sin A)/(1 + sin A) • Use identities: 1 − sin^2 A = cos^2 A • Assume 1 + sin A ≠ 0 so the expression is defined

Concept / Approach: Multiply by the conjugate (1 − sin A)/(1 − sin A): (1 − sin A)/(1 + sin A) = (1 − sin A)^2 / (1 − sin^2 A). Then use 1 − sin^2 A = cos^2 A. Finally rewrite (1 − sin A)/cos A as sec A − tan A, and square it.

Step-by-Step Solution: 1) Start: (1 − sin A)/(1 + sin A) 2) Multiply numerator and denominator by (1 − sin A): = (1 − sin A)^2 / ((1 + sin A)(1 − sin A)) 3) Denominator becomes a difference of squares: (1 + sin A)(1 − sin A) = 1 − sin^2 A 4) Use identity: 1 − sin^2 A = cos^2 A 5) So: (1 − sin A)/(1 + sin A) = (1 − sin A)^2 / cos^2 A 6) Separate as a square: = ((1 − sin A)/cos A)^2 7) Rewrite: (1 − sin A)/cos A = 1/cos A − sin A/cos A = sec A − tan A 8) Final: = (sec A − tan A)^2

Verification / Alternative check: Take A = 30°: sin A = 1/2, cos A = √3/2. LHS = (1 − 1/2)/(1 + 1/2) = (1/2)/(3/2) = 1/3. RHS: sec A − tan A = (2/√3) − (1/√3) = 1/√3, square gives 1/3. Matches.

Why Other Options Are Wrong: • sec A − tan A (not squared) is not equal to the original ratio. • sec A + tan A corresponds to a different identity. • tan A or sec A alone cannot match the rationalized squared form.

Common Pitfalls: • Forgetting to square after converting to (1 − sin A)^2/cos^2 A. • Mistakenly using 1 − sin^2 A = 1 − cos^2 A.

Final Answer: (sec A − tan A)^2